Avrami model nucleation

A kinetic model for single-phase polymerizations— that is, reactions where because of the similarity of structure the polymer grows as a solid-state solution in the monomer crystal without phase separation—has been proposed by Baughman [294] to explain the experimental behavior observed in the temperature- or light-induced polymerization of substimted diacetylenes R—C=C—C=C—R. The basic feature of the model is that the rate constant for nucleation is assumed to depend on the fraction of converted monomer x(f) and is not constant like it is assumed in the Avrami model discussed above. The rate of the solid-state polymerization is given by... 

Applying the Avrami model to the analysis of the isothermal crystallization of interesterified and noninteresterified 20%SSS/80%000 at 30°C, 40 C and 50 C, many differences can be observed (Table 17.3). At 30°C and 40°C growth would be described as rodlike with instantaneous nucleation for both interesterified and noninteresterified samples. Also, for the noninteresterified system at 50°C spherulitic growth with instantaneous nucleation takes place. The half-time of nucleation... 

It was often found that, contrary to the theoretical prediction, the value of n is non-integer [Avrami, 1939]. The Avrami model is based on several assumptions, such as constancy in shape of the growing crystal, constant rate of radial growth, lack of induction time, uniqueness of the nucleation mode, complete crystallinity of the sample, random distribution of nuclei, constant value of radial density, primary nucleation process (no secondary... 

The kinetics of crystallization has been simulated by many models, including the Avrami model (House 2007). The rate of conversion from amorphous to crystalline states can be measured by using thermal analysis (differential scanning calorimetry, DSC) and/or X-ray diffraction. The rate of conversion from amorphous to crystalline form depends on a number of factors. The process occurs in two steps, nucleation and growth (Mullin 2001), which are affected by various factors and occur at different rates. Specifically, for crystallization to occur, a seed or nucleus must form, on which subsequent growth will occur. Thus, the rate of nucleation is of primary interest. By analogy with Arrhenius-type processes, the nucleation rate can be written as... 

This form of the Avrami equation describes a uniform 3-d growth of spherical particles whose nuclei randomly form at a constant average rate per unconverted volume, and is of a mathematical form referred to as the Avrami (A4) nucleation model (Khawam and Flanagan 2006). Other forms for other growth patterns have been derived, such as for crystals that grow as disks as (House 2007)... 

The ciystallization curve shown in Figure 2 was obtained in a SAXS/WAXS/DSC experiment from iPP [23] and shows the classic features of primary crystallization. The detailed molewlar structure of the polymer, the specific nature of the nucleation processes and the degree of under-coolteg, determines the magnitude of the lamellar thickness and the degree of crystallinity within the lamellar stadcs. The crystallization kinetics are analyzed using the Avrami model [24], expressed in terms of the equation... 

Actually, it is very rare to obtain integer values of the Avrami exponent, which suggests the existence of parallel and/or competing processes of nucleation and growth of the crystalline zones. The treatment of experimental data using the Avrami model has thus less significance. Other, more complex models, which are better adapted to the case of polymers, were proposed—in particular, those that take into account the necessary disentanglement of the chains before crystallization. 

The modified Avrami model was derived for the case of three-dimensional random nucleation followed by uniform linear growth of the nuclei. In its linearized form. 

Simulation of phase transformations in these examples have some predictive capability. Fot oomposile systems, die Avrami model as such, is claimed not to apply because nucleation is eoKtrained by fbo- loading and sui oe sites. For instance, factors such as fiber fraction and diameter, spherulitic growth (normal and transcrystalline) must all figure in the overall composite morphology. 

Fig. 6. Two reaction models which result in obedience to the power law [eqn. (2), n = 2 ] at low a, or the Avrami—Erofe ev equation [eqn. (6), n = 2 ] over a more extensive range of a. In (a), there is growth of semi-circular nuclei in a thin plate of reactant in (b), there is cylindrical growth of linear internal nuclei. In both examples, rapid nucleation (0 = 0) is followed by two-dimensional growth (X = 2).

Under hydrothermal conditions (150-180 °C) maghemite transforms to hematite via solution probably by a dissolution/reprecipitation mechanism (Swaddle Olt-mann, 1980 Blesa Matijevic, 1989). In water, the small, cubic crystals of maghemite were replaced by much larger hematite rhombohedra (up to 0.3 Lim across). Large hematite plates up to 5 Lim across were produced in KOH. The reaction conditions influenced both the extent of nucleation and crystal morphology. The transformation curve was sigmoidal and the kinetic data in water and in KOH fitted a first order, random nucleation model (Avrami-Erofejev), i.e. 

Ball (31) studied the solid-state hydrolysis of single crystals of aspirin, and also observed S-shaped plots of fraction decomposed vs. time, but found that the kinetic data fit an Avrami-Erofeyev model involving nucleation at dislocations in the crystal lattice. 

The different combinations of nucleation, growth, and impingement processes give rise to the Johnson-Mehl-Avrami kinetic model [4], which results in the following equation... 

For a solid-state reaction, one of the solutions of Equation 3.1 is the Avrami-Erofeev equation [3], The phase transition model that derives this equation supposes that the germ nuclei of the new phase are distributed randomly within the solid following a nucleation event, grains grow throughout the old phase until the transformation is complete. Then, the Avrami-Erofeev equation is [3]... 

Nucleation and growth processes such as glass crystallization generally follow the Johnson-Mehl-Avrami (JMA) model ... 

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